THE SUPPORT GENUS OF CERTAIN LEGENDRIAN KNOTS

YOULIN LI AND JIAJUN WANG Abstract. In this paper, the support genus of all Legendrian right handed trefoil knots and some other Legendrian knots is computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genus with arbitrarily negative Thurston-Benniquin invariant. This answers a question in [O09].

arXiv:1101.5213v1 [math.GT] 27 Jan 2011

1. Introduction In his seminal paper [G02], Giroux established the surprising one-to-one correspondence between the contact structures up to isotopy and the open book decompositions up to positive stabilizations on a given closed oriented three-manifold. See [E06] for details. It becomes natural and convenient for topologists to study contact structures in the viewpoint of open book decompositions. For Legendrian knots, Akbulut and Ozbagci [AO01] showed that for any Legendrian link L in S 3 with the standard contact structure std , there exists a compatible open book such that L sits in a page of the open book, furthermore, the framing of L given by the page of the open book agrees with the contact framing. See also [P04]. In [EO08], Etnyre and Ozbagci introduced the denition of support genus for a contact threemanifold, which is the minimal genus of a page among all open books supporting the given the contact three-manifold. In a similar fashion, Onaran dened in [O09] an invariant for Legendrian knots in a contact three-manifold as follows. Denition 1.1 ([O09]). Let L be a Legendrian knot in a contact three-manifold (M, ), the support genus of L, denoted by sg(L), is the minimal genus of a page among all open book decompositions of M supporting such that L sits on a page of the open book and the framings given by and given by the page coincide. Ding and Geiges [DG04] introduced the denition of contact surgeries along a Legendrian link. If a contact three-manifold (M, ) is obtained by a contact r surgery along a Legendrian knot L in (S 3 , std ), Onaran showed that sg(L) is greater than or equal to the support genus of (M, ) ([O09, Remark 5.11]). In the same paper, the following question is asked: Question 1.2. Does every Legendrian knot in (S 3 , std ) with negative Thurstion-Bennequin invariant have support genus zero? In the present paper, we study the support genus of certain Legendrian knots in (S 3 , std ). First, for Legendrian torus knots, we have the following Theorem 1.3. Suppose k 1. Let L be a Legendrian torus knot T (2, 2k + 1) in (S 3 , std ) with nonnegative Thurston-Bennequin invariant, then sg(L) = 1. Next, we study the support genus of Legendrian twist knots K2m , where m 1. (See [ENV10] for the meaning of K2m .) In particular, K2 is the right handed trefoil. All Legendrian twist knots are classied in [ENV10]. In fact, K2m has m2 /2 Legendrian representatives with ThurstonBennequin invariant one and rotation number zero, and has a unique Legendrian representative with Thurston-Bennequin invariant minus one and rotation number zero ([ENV10, Theorem 1.1 (4)]). We have the following1

YOULIN LI AND JIAJUN WANG

Theorem 1.4. Suppose m, n1 and n2 are natural numbers. Let L be a Legendrian representative of n n the twist knot K2m in (S 3 , std ) with Thurston-Bennequin invariant 1. Then sg(S+1 S2 (L)) = 0, where S+ and S denote the positive and negative stabilizations respectively. Theorems 1.3 and 1.4 both computed the support genus of some Legendrian right-handed trefoil knots. By the classication results in [EH01], the remaining Legendrian right-handed trefoil knots n n are S+ (L) and S (L) for n 2. The support genus of these Legendrian knots is computed in the following theorem, and thus we computed the support genus of all Legendrian right-handed trefoil knots. Theorem 1.5. Let L be a Legendrian right handed trefoil knot in (S 3 , std ) with Thurston-Bennequin n n invariant 1. Then for any integer n 2, both S+ (L) and S (L) have support genus 1. This gives a negative answer to Question 1.2. In fact, Legendrian knots with positive support genus can have arbitrarily negative Thurston-Bennequin invariant. Our strategy is the following. We will construct a bered link with a Thurston norm minimizing Seifert surface which contains the interested knot. By computing the Thurston-Bennequin invariant and rotation number of the Legendrian realization of this knot and known classication results, we will be able to determine the Legendrian knot and get an upper bound for the support genus for it. In the other direction, we use the Heegaard Floer contact invariant to get lower bound for the support genus. Combining these results, we will compute the support genus of our interested Legendrian knots. Acknowledgement. Part of the work was done when the rst named author was visiting Peking University. He would like to thank the School of Mathematical Sciences in Peking University for their hospitality. He is also partially supported by NSFC grant 11001171. 2. Proof of Theorem 1.3 The surface F illustrated in Figure 1 is a punctured torus and contains an embedded torus knot T (2, 2k + 1) in its interior.

Figure 1. A punctured torus containing the torus knot T (2, 2k + 1). As illustrated in Figure 2, F can be obtained by a sequence of positive stabilizations from the disk. By [G02], F can be viewed as a page of an open book decomposition of S 3 which supports the standard tight contact structure std . K can be realized as a Legendrian knot since K is

THE SUPPORT GENUS OF CERTAIN LEGENDRIAN KNOTS

homologically nontrivial (in fact, nonseparating) in F , which we denote by Tm (2, 2k +1). Moreover, the contact framing of Tm (2, 2k + 1) coincides with the page framing induced by F . This is because the contact planes can be arranged to be arbitrarily close to the tangent planes of the pages. Let J be a push-o of Tm (2, 2k + 1) along the surface F , then J represents the contact framing of Tm (2, 2k + 1). The Thurston-Bennequin invariant of Tm (2, 2k + 1) equals to the linking number of Tm (2, 2k + 1) and J, which is easily seen to be 2k 1. By the classication of Legendrian representatives of torus knots in [EH01], Tm (2, 2k + 1) has the maximum Thurston-Bennequin invariant over all Legendrian representatives of T (2, 2k + 1) and it is the unique one. So the support genus of the Legendrian knot Tm (2, 2k + 1) is at most 1. According to [EH01], all other Legendrian representatives of the torus knot T (2, 2k + 1) can be obtained by stabilizing Tm (2, 2k + 1). Thus, by [O09, Theorem 5.9], the support genus of any Legendrian representative of T (2, 2k + 1) is at most 1. On the other hand, since a Legendrian knot in a weakly llable contact structure with positive Thurston-Bennequin invariant has positive support genus ([O09, Lemma 5.4]), the support genus of Tm (2, 2k + 1) is at least 1. Therefore the support genus of Tm (2, 2k + 1) is 1. This ends the proof of Theorem 1.3.

YOULIN LI AND JIAJUN WANG

3. Proof of Theorem 1.4 By [O09, Theorem 5.9], it suces to show that the Legendrian representative of the twist knot K2m with Thurston-Bennequin invariant minus one and rotation number zero has support genus zero. The punctured sphere illustratd in Figure 3 contains the twist knot K2m in its interior. It is

Figure 3. A punctured sphere containing the twist knot K2m . easy to see that F is a page of an open book decomposition which corresponds to (S 3 , std ). Since K is not null homologous in F , K can be made Legendrian and the contact framing of K coincides with the page framing induced by F . The linking number of K and its push-o along F is 1, by the same argument as in the previous section, the Thurston-Bennequin invariant of K is 1. Below we shall compute the rotation number of K. We turn the open book decomposition shown in Figure 3 into an abstract open book decomposition (F , t1 t2 . . . tm+2 ) shown in Figure 4, where F is a punctured sphere which is homeomorphic to F , and ti denotes the right handed Dehn twist along i , i = 1, 2, . . . , m + 2.

Figure 4. An abstract open book. If we perform a Legendrian surgery along K, then we obtain a Stein llable tight contact manifold which corresponds to the planar open book decomposition (F , t1 t2 . . . tm+2 tK ). Let (W, J) be the Stein surface obtained from B 4 by attaching a two-handle corresponding to the Legendrian

THE SUPPORT GENUS OF CERTAIN LEGENDRIAN KNOTS

surgery along K. Let c1 (J) be the rst Chern class of this Stein surface, and h be the generator of H2 (W ; Z) Z supported on the attached 2-handle, then c1 (J), h = rot(K). = The open book decomposition in Figure 4 is the same as the one in Figure 5, where each upper

Figure 5. Another diagram for the abstract open book. horizontal segment is identied with the corresponding lower horizontal segment to form a onehandle. We consider the Kirby diagram in Figure 6, where the framings are labelled with respect to

Figure 6. A Kirby diagram for the Stein surface. the blackboard framings. According to [E90], this Kirby diagram presents a Stein surface, denoted by (W , J ). By handle cancellation, it is not hard to see that W and W are dieomorphic. Since the induced contact structures on W by (W, J) and (W , J ) are isotopic, we have c1 (J) = c1 (J ) by [LM97, Theorem 1.2].

THE SUPPORT GENUS OF CERTAIN LEGENDRIAN KNOTS

3 3 are distinct primitive elements in HF (Sn+1 (T (2, 3))). Each c(i ) lies in HF (Sn+1 (T (2, 3)), sj ) for some j = 0, 1, . . . , n. On the other hand, there are exactly two distinct primitive elements 3 in Z(T + ) and there are exactly n + 1 Z(T + ) summands in HF (Sn+1 (T (2, 3))). Moreover, also by Theorem 4 in [P04], for i1 = i2 , c(i1 ) and c(i2 ) cannot both belong to a Z(T + ) summand. So at least 3 one of c(i ), i = 1, 2, . . . , n+2, does not belong to the n+1 Z(T + ) summands in HF (Sn+1 (T (2, 3))). We denote this element as c(i0 ). Therefore, c+ (i0 ) does not belong to the n + 1 T + -summands 3 3 in HF + (Sn+1 (T (2, 3))). Hence c+ (i0 ) does not vanish in HFred (Sn+1 (T (2, 3))) Z. = By Theorem 1.2 in [OSSz05], the contact manifold i0 does not admit a planar open book decomposition. By Theorem 5.10 in [O09], the support genus of Li0 is positive. When 1 n 6, by Theorem 5.9 in [O09], among the Legendrian right handed trefoil knots with Thurston Bennequin invariant n, there must be one whose support genus is positive. Let L be a Legendrian right handed trefoil with Thurston-Bennequin invariant 1, then, by n n Theorem 1.3, the support genus of S+1 S2 (L) is 0, where ni 1 (i = 1, 2). On the other hand, for n n n 2, the support genus of S+ (L) and S (L) are equal, since they only dier in the orientation. n (L) and S n (L) are 1. So the support genus of both S+